1 kg particle at a height of 4 m has a speed of 2 m/s down a incline making an angle `53^@` with horizontal as shown in figure. It slides on a horizontal section of length 3 m at ground level and then up an incline making an angle `37^@` with horizontal. All surfaces have `mu_k=0.5`. How far from point O (bottom of right inclined plane), along the incline making and angle `37^@` with horizontal, does the particle first come to rest?

Explanation

Let x be the distance travelled by the particle along the `37^@` incline, after which it comes to rest. From work-energy thorem, work done by gravity on block on incline `53^@`
`+` work done by friction on block on incline `53^@`
`+` work done by friction on ground level `+` work done by gravity on block on incline `37^@+` work done by friction on block on incline `37^@=`change in K.E.
`implies=mg(4)-mumgcos53^@(5)-mumgcos37^@(x)-mgsin37^@(x)=0-(1)/(2)(1)(2)^2`
`implies40-0.5xx10xx(3)/(5)xx5-0.5xx10xx3`
`-0.5xx10xx(4)/(5)(x)-(3)/(5)(x)=0-2`
On solving we get `x=1.2`m